1. Central question for the day
How do we give dimension and meaning to our descriptions of the world around us?

2. Measurement, Uncertainty and Significant Figures

3. Introduction to derived units
Why are all measurements considered to be arbitrary?
Why do we make measurements?
What are derived units?

4. Measurement and Uncertainty
All measurements have uncertainty because...
All measurement involves estimation;
The precision of the instruments limits our ability to estimate.
When reporting a measured value,
Include all certain digits and ONE estimated digit.
All certain digits plus one estimated digit are called significant digits or significant figures.

5. Reading Scales
Uncertainty – Limit of precision of the reading (based on the precision of the scale and your ability to estimate the final digit).

6. How long is it? (1) Left: Right:
1.4 in +0.1 in
Right:
1.47 in in
For each of the rulers, give the correct length measurement for the steel pellet. Indicate uncertainty by adding +.

7. Read thermometers in °C
(2)
(3)
1.55 °C °C
(4)
30.0 °C °C
19.3 °C °C

8. Reading Graduated Cylinders
Use the BOTTOM of the meniscus (the curved interface between air and liquid) as a point of reference in making measurements of volume in a graduated cylinder, pipet, or buret.
In reading any scale, your line of sight should be perpendicular to the scale to avoid 'parallax' reading errors.
5.73 mL mL
(5)

9. (6)
(7)
0.35 mL mL
3.0 mL mL

10. Measured Values vs. Numbers
Science is based on measurement
All measurements have
Magnitude
Units
Uncertainty
Mathematics is based on numbers
Exact numbers are obtained by
Definition
Counting
No Naked Numbers in Science!!!

11. Counting numbers
Numbers obtained by counting have no uncertainty unless the count is very large.
For example, the word 'sesquipedalian' has 14 letters. You can count them.
"14 letters" is not a measurement, since that would imply that we were uncertain about the count in the ones place.
14 is an exact number here.

12. Counting numbers
Very large counts often do have some uncertainty in them, because of inherent flaws in the counting process or because the count fluctuates.
For example, the number of human beings in Arizona would be considered a measurement because it can not be determined exactly at the present time.

13. Numbers from definitions
Numbers obtained from definitions have no uncertainty unless they have been rounded off.
For example, a foot is exactly 12 inches. The 12 is not uncertain at all.
A foot is also exactly centimeters from the definition of the centimeter. The 8 in is not uncertain at all.
But if you say 1 foot is 30.5 centimeters, you've rounded off the definition and the rounded digit is uncertain.

14. Practice
Which of the following quantities can be determined exactly? (Select all that are NOT measurements.)
The number of light switches in the room you're sitting in now
The number of ounces in one pound
The number of stars in the sky
The number of inches per meter
The number of red blood cells in exactly one quart of blood

15. Significant Figures/Digits
All reported measurements should include ONE and only one estimated digit.
All the certain digits plus the one estimated digit are called significant digits.
Example: A sample of liquid has a measured volume of mL.
1. Which digits are certain and which digit is estimated?
2,3,0 are certain; 1 is estimated
2. How many significant digits does the measurement have? 4
3. How far apart are the closest markings on the cylinder, in mL?
0.1 mL

16. Significant Figures Practice
Consider the following measurements. The estimated digit is in red:
Measurement Number of Distance Significant Digits Between Markings on Measuring Device
142.7 g __ __ g
103 nm __ __ nm
x 108 m __ _______ m

17. Significant Figures Practice
Consider the following measurements. The estimated digit is in red:
Measurement Number of Distance Significant Digits Between Markings on Measuring Device
142.7 g g
103 nm nm
x 108 m x 108 m

18. Counting Significant Digits
Consider a sample has a measured mass of g.
124.1 g can also be expressed as kg
By expressing the measurement in different units, is the estimated digit changed?
Should these two values have the same number of sig. fig.?
Moving the decimal place doesn't change the fact that this measurement has 4 significant figures.
No.
Yes.

20. Sig. fig. rounding and more
Central idea: When using scientific notation and sig figs what should it look like?

21. Counting Significant Digits
Suppose a mass is given as 127 ng.
That's µg, or mg, or g.
These are all just different ways of writing the same measurement. So all have the same number of significant digits: 3.
Should the leading zeros be counted as sig. fig.?
Leading zeros are not counted as sig. fig. because they are place holders (holding where the decimal is).

22. Counting Significant Digits
Determine the number of significant digits in the following series of numbers:
kg = g = 341 mg
12 µg = g = kg
Mg = kg = g
Is the last zero in g significant?
No, because should only have 4 sig. fig., same as Mg and kg.

23. Are all trailing zeros not sig. fig.?
km has ____ sig figs because the last zero on the right is the ________ digit so it is _______.
(3 sig figs; estimated; significant)
What about 150 m?
If measured by this ruler, which digit is estimated? How many sig fig?
100
200
5 is estimated; 2 sig fig
If measured by this ruler, which digit is estimated? How many sig fig?
200
100
0 is estimated; 3 sig fig

24. Trailing Zeros, cont’d
Without knowing the uncertainty of the measuring device, we are not sure if the reported value of 150 m has 2 or 3 sig. figs.
Therefore, we can not assume the trailing zero is significant.
We can only say it has at least 2 sig. figs.
Conclusion: if there is a decimal, trailing zeros are sig figs. If there is no decimal, trailing zeros are not assumed as sig figs.

26. Counting Significant Digitsm
3 sig figs m
4 sig figs
32.00 mL
4sig figs
The leading zeros are not significant but the trailing zero are when there is a decimal.
321,210,000 miles
5 sig figs (at least)
84,000 mg
2 sig figs (at least)
The trailing zeros can not be assumed significant when there is no decimal.

27. What about zeros in the middle?
lbs
6 sig figs
Zeros sandwiched between two significant digits are always significant.

28. Let’s summarize: When are zeros significant?
Determine the number of significant figure for measurements in each group and state the rule for counting sig. fig. in each group.
km kg Mg
5 SF 4 SF 6 SF
Rule ___________________________________
m mm µm
1 SF 3 SF 5 SF

29. When are zeros significant?
m mm µm
2 SF 5 SF 5 SF
Rule ___________________________________
3000 m µm ,900,000 miles
at least: 1 SF SF 3 SF

30. A better way for trailing zeros…
To avoid the ambiguity about the trailing zeros, it is always better to record a large number in scientific notation.
For example, if 3000 m was measured to the nearest meter (last zero is estimated), the measurement should be written as
3.000 x 103 m

31. Scientific Notation
When reporting very small or very large values, it is convenient to use scientific notation.
L = 3.4 x 10-3 L
g = x 104 g
The power of 10 equals to the number of decimal place being moved.
True scientific notation can only have one non-zero digit before the decimal.
0.34 x 10-2 L and g x 103 g are not true scientific notations.

32. Scientific Notation and Sig Figs
When converting a measurement to scientific notation, the number of sig figs should not be changed.
Example: kg
4 SF x 10-2 kg kg
5 SF x 10-2 kg

33. Practice
How many significant figures are there in each of the following measurements? g g g g g
100 g
Express the above measurements using scientific notation.

34. Answers to Practice
How many significant figures are there in each of the following measurements? SF
g x 103
g x 104
g x 10-3
g x 10-2
g at least x 105
100 g at least 1 1 x 102